# 4.11 Parameters for a Binomial Distribution

Kanya Shah

In order to use binomial formulae for mean and standard deviation, you need to identify whether the conditions qualify for binomial distribution. If one of four conditions don’t work/occur, it is not a binomial setting. In other words, you can’t just use binomial distribution because you feel like it. 10% Condition: When taking a random sample of size n from a population of size N. we can use a binomial distribution to model the count of successes in the sample as long as n < 0.10N.
The four conditions for a binomial setting are Binary, Independent, Number, and Same Probability or BINS. Remember your BINS for the AP Stats exam questions.
• Binary: The possible outcomes of each trial can be classified as a success or a failure.
• Independent: Trials must be independent. That is, knowing the outcome of one trial must not tell us anything about the outcome of any other trial.
• Number: The number of trials n of the chance process must be fixed in advance.
• Same Probability: There is the same probability of success p on each trial.

## Describing Mean and Standard Deviation of Binomial Variables

Disclaimer: These formulae only work for binomial settings!
To find the mean of a binomial random variable: If a count X of successes has a binomial distribution with number of trials n and probability of success p, the mean (expected value) of X is mean = n*P.
To find the standard deviation of a binomial random variable: If a count X of successes has a binomial distribution with number of trials n and probability of success p, the standard deviation of X is standard deviation = np(1-p)

## Binomial Distributions in Statistical Sampling:

10% Condition: When taking a random sample of size n from a population of size N. we can use a binomial distribution to model the count of successes in the sample as long as n < 0.10N.

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